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Behavior of Arbitrarily Shaped Ce2O3 Clusters at the Ar Gas/Liquid Steel Interface

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Abstract

Understanding inclusions’ behavior at the gas/steel and slag/steel interfaces is essential for inclusion removal and clean steel production. The agglomeration of inclusions into clusters at the gas/liquid steel interface was frequently observed in situ using confocal scanning laser microscopy. For the sake of simplicity, non-spherical inclusions/clusters are usually simplified as spheres with an equivalent radius. Therefore, the meniscus around a single inclusion becomes rotationally symmetric, and the corresponding capillary force can be easily derived. However, the meniscus and the pairwise capillary interaction are heavily dependent on particle shape. In this work, the recently developed sub-particle model is applied to quantitatively describe the meniscus shape and the capillary force between arbitrarily shaped Ce2O3 clusters accounting fully for their shape. In the capillary force calculation, the resistive drag force on Ce2O3 clusters is considered when they are sliding on the liquid steel surface. The result shows that the sub-particle model predicts the capillary force better than the simplified spheres compared to the experimental data.

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Acknowledgments

The authors thank the China Scholarship Council (CSC) for financial support (File No. 201706080018).

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On behalf of all authors, the corresponding author states that there is no conflict of interest.

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Appendix

Appendix

Moving Average (MA) Algorithm

Since the measured inter-inclusion distance errors are inevitable, the moving average (MA) algorithm is applied to reduce the effect of the system errors. In this algorithm, the number of data points to be used for the averaging is an important factor. This Appendix will quantitatively evaluate the effect of the numbers of averaged data points on the accuracy of calculated force. The raw distance data are derived from a simple simulation, in which two 10 µm spherical quadrupolar particles are driven by the pairwise capillary force from a distance of 80 µm. The capillary force can be calculated from Eq. [6] in the main text. In the calculation, the physical properties of particles and liquids are the same as used in the main text. In addition, the angle hysteresis is 1 deg and the phase shift is 0 deg; the time interval for data recording is 0.033 s. During the particle-approaching process, there is also a resistive (drag) force on the particles from the liquid phase, and this force can be calculated from reference [33]. This is an imitation of spherical inclusions moving on the molten steel surface. The separation distance as a function of time is shown in Figure A1 (solid blue line), and the corresponding forces on particles are shown in the insert. It is interesting to note that the net force on particles is much smaller than the drag force or the capillary force. Gaussian noise with a standard deviation of 0.7 µm (dimension of a pixel in the CSLM experiment) is added to the raw distance data to simulate the effect of the system errors, see the red circles in Figure A1.

Fig. A1
figure 11

The separation distance as a function of time with and without Gaussian noise. Insert: The forces on particles as a function of time

We will test the effect of the number (k) of data points for the averaging from two data points (MA2) to twelve data points (MA12). The demonstration of the averaged distance data points using four, eight, and twelve data points is shown in Figure A2(a). The more the data points are used in the averaging, the more deviation from the original data is obtained. The corresponding velocity calculated from Eq. [12] in the main text is shown in Figure A2(b). The insert confirms the validity of the method used to calculate particles’ velocity, where the original distance data are used. As can be seen, the oscillation of particle velocity at a long distance is suppressed when increasing the number of data points for averaging.

Fig. A2
figure 12

(a) The separation distance as a function of time with raw data and averaged data. (b) The velocities as a function of distance calculated with and without data averaging; insert: the velocity calculated from the raw distance data

The sum of squared residuals (SSR) between the simulated velocity and the velocities derived from the averaged distance is shown in Figure A3. The result shows that a larger number of data points used in the averaging algorithm cannot guarantee a more accurate velocity in this case. Three or four data points for the averaging are good enough to reduce the effect of the system errors.

Fig. A3
figure 13

The SSR as a function of the number of data points used in the averaging

Effect of distance smoothing

To study the effect of the distance soothing on the attractive force calculation, we compared the force calculated from the raw data and the smoothed data, as shown in Figure A4. The smoothing is performed on three successive data points using Eq. [7], and the attractive force is calculated from Eqs. [8] through Eq. [12].

Fig. A4
figure 14

Attractive force calculated from the CSLM experiment with and without distance smoothening. MA3 represents the raw data that are smoothed with three successive raw data points using the moving average algorithm

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Qiu, Z., Malfliet, A., Blanpain, B. et al. Behavior of Arbitrarily Shaped Ce2O3 Clusters at the Ar Gas/Liquid Steel Interface. Metall Mater Trans B 53, 3896–3908 (2022). https://doi.org/10.1007/s11663-022-02650-y

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